Optimal dynamic scale and structure of a
multi-pollution economy
University of Lüneburg
Working Paper Series in Economics
No. 50
May 2007
www.uni-lueneburg.de/vwl/papers
ISSN 1860 - 5508
von
Optimal dynamic scale and structure
of a multi-pollution economy
Stefan Baumg¨artnera, Frank J¨ostb and Ralph Winklerc, ∗
a Centre for Sustainability Management, Leuphana University of L¨uneburg,
Ger-many
b Alfred-Weber-Institute of Economics, University of Heidelberg, Germany c Center of Economic Research, ETH Swiss Federal Institute of Technology, Zurich,
Switzerland
Abstract: We analyze the optimal dynamic scale and structure of a two-sector-economy, where each sector produces one consumption good and one specific pol-lutant. Both pollutants accumulate at different rates to stocks which damage the natural environment. This acts as a dynamic driving force for the economy. Our analysis shows that along the optimal time-path (i) the overall scale of eco-nomic activity may be less than maximal; (ii) the time scale of ecoeco-nomic dynamics (change of scale and structure) is mainly determined by the lifetime of pollutants, their harmfulness and the discount rate; and (iii) the optimal control of economic scale and structure may be non-monotonic. These results raise important ques-tions about the optimal design of environmental policies.
Keywords: dynamic economy-environment interaction, multi-pollutant emis-sions, non-monotonic control, optimal scale, stock pollution, structural change, time scale
JEL-classification: Q20, O10, O41
Correspondence:
Stefan Baumg¨artner, Leuphana University of L¨uneburg, Centre for Sustainability Management, P.O. Box 2440, D-21314 L¨uneburg, Germany, phone: +49.4131.677-2600, fax: +49.4131.677-1381, e-mail: [email protected]
∗
We are grateful to Ulf Moslener, John Proops, Martin Quaas, Till Requate; seminar par-ticipants at Ascona (SURED 2004), Heidelberg and Budapest (EAERE 2004). We thank the Simulation and Optimization Group of the Interdisciplinary Center for Scientific Computing
1
Introduction
The natural environment is being damaged by the stocks of various pollutants, which are produced in different sectors of the economy, accumulate according to different dynamic relationships, and damage different environmental goods. As an example, think of the two economic sectors ‘agriculture’ and ‘industry’. Nitrate and pesticide run-off from agricultural cultivation accumulates in groundwater and decreases its quality as drinking water (UNEP 2002); carbon dioxide emissions from fossil fuel combustion in the industrial sector accumulate in the atmosphere and contribute to global climate change (IPCC 2001). In general, the different pollutants differ in their internal dynamics, i.e. natural degradation processes, and in their harmfulness. This has implications for the optimal dynamics of both the scale and structure of the economy. By scale we mean the overall level of economic activity, measured by total factor input; by structure we mean the composition of economic activity, measured by relative factor inputs to different sectors.
In this paper, we look into these coupled environmental-economic dynamics from a macroeconomic point of view. In particular, we are interested in the fol-lowing questions: How should the macroeconomic scale and structure change over time in response to the dynamics of environmental pollution? Is this dynamic pro-cess monotonic over time, or can a trade-off between long-run and short-run con-siderations (e.g. lifetime versus harmfulness of pollutants) induce a non-monotonic economic dynamics? What is the time scale of economic dynamics (i.e. change of scale and structure), and how is it influenced by the different time scales and constraints of the economic and environmental systems? These questions are rel-evant for the current policy discussion on the sustainable biophysical scale of the aggregate economy relative to the surrounding natural environment (e.g. Arrow et al. 1995, Daly 1992, 1996, 1999), and how economic policy should promote struc-tural economic change as a response to changing environmental pressures (e.g. de
Bruyn 1997, Winkler 2005).
We address these questions based on a model which comprises two economic sectors, each of which produces one distinct consumption good and, at the same time, gives rise to one specific pollutant. Both pollutants accumulate to stocks which display different internal dynamics, in the sense that the respective natu-ral deterioration rates differ, and cause welfare decreasing environmental damage independently of each other. Of course, this relatively simple model cannot offer detailed policy prescriptions. However, it is detailed enough to clarify the underly-ing theoretical issues. In fact, we perform a total analysis of economy-environment interactions in a twofold manner. First, we analyze a multi-sector economy, which is fully specified in terms of resource endowment, technology, preferences and en-vironmental quality. Second, we consider a ‘disaggregate’ natural environment. This goes beyond many contributions to environmental economics, where either only one (aggregate) pollutant is considered or different pollutants give rise to the same environmental problem.
Many studies in the extant literature assume that it is the flow of emissions which causes environmental problems. This neglects stock accumulation and, thus, an essential dynamic environmental constraint on economic action. Stock pollution has been taken into account by some authors (e.g. Falk and Mendelsohn 1993, Forster 1973, Luptacik and Schubert 1982, Van der Ploeg and Withagen 1991). This is usually done at a highly aggregated level, such that only one pollutant is taken into account. The case of several stock pollutants which all contribute to the same environmental problem (climate change) has been studied by Michaelis (1992, 1999). He is interested in finding cost-effective climate policy measures in the multi-pollution case for a given structure of the economy and does not explicitly consider the dynamics of the production side of the economy. Aaheim (1999) goes beyond Michaelis in that he analyzes numerically the dynamics of a
is constrained by an exogenously given policy target concerning the aggregate level of pollution. Moslener and Requate (2001) challenge the global warming potential as a useful indicator when there are many interacting greenhouse gases with different dynamic characteristics. Faber and Proops (1998: chap. 11) and Keeler et al. (1972) explicitly study the dynamics of different production sectors with pollution, assuming one single pollutant. Winkler (2005) analyzes optimal structural change of a two-sector economy characterized by two stock quantities: the capital stock and the stock of a pollutant which is emitted from the more capital-intense sector. Baumg¨artner and J¨ost (2000) study the optimal (static) structure of a vertically integrated two-sector economy where both sectors produce a specific by-product. The first sector’s by-product can be used as a secondary resource in the second sector.
In this paper, we determine the optimal dynamic scale and structure of a multi-pollution economy within an optimal control framework. We use a linear approximation around the steady-state to obtain analytical results, and a numer-ical optimization of the non-approximated system to check for their robustness. The methodological innovation of our analysis is that we derive a closed form solution to the intertemporal optimization problem, which includes explicit ex-pressions for the time scale of economic dynamics and the point in time where a non-monotonicity may occur. Our analysis shows that along the optimal time-path (i) the overall scale of economic activity may be less than maximal; (ii) the time scale of economic dynamics is mainly determined by the lifetime of pollu-tants, their harmfulness and the discount rate; and (iii) the control of economic scale and structure may be non-monotonic.
Although our modeling approach is inspired by Ramsey-type optimal growth models, which have previously been used to study steady state growth with envi-ronmental pollution (e.g. Gradus and Smulders 1993, 1996, J¨ost et al. 2004, Keeler et al. 1972, Plourde 1972, Siebert 2004, Smith 1977, Van der Ploeg and Withagen
1991), we are essentially concerned with the issue of dynamic change in both scale and structure of economic activity. Therefore, in this paper we do not restrict the analysis to steady states but focus on the explicit time-dependence of the solu-tion. Furthermore, we study an economy without any potential for steady state growth, as this highlights the structural-change-effect, which may be obscured by growth effects otherwise. The sole genuine generator of dynamics in our model is the accumulation of pollutant stocks in the natural environment.
The paper is organized as follows. In Section 2 we present the model. Section 3 is devoted to a formal analysis of the optimal dynamic scale and structure of the economy, based on a linear approximation around the stationary state. Section 4 confirms the analytical results thus obtained by a numerical optimization of the non-approximated system. Section 5 concludes.
2
The model
We study a two sector economy with one scarce non-accumulating factor of pro-duction, say labor, two consumption goods, and two pollutants that accumulate to stocks. Welfare is determined by the amounts consumed of both consumption goods, as well as by the environmental damage caused by the two pollutant stocks. The production of consumption goods in sectors 1 and 2 of the economy is described by two production functions, yi = Pi(li) for i = 1, 2, where li denotes
the amount of labor allocated in sector i. With index l denoting derivatives with respect to the sole argument li, Pli ≡ dPi/dli and Plli ≡ d2Pi/dli2, the production
functions are assumed to exhibit the following standard properties: Pi(0) = 0 , Pli> 0 , lim
li→0
Pli= +∞ , Plli < 0 (i = 1, 2) . (1) Since we want to analyze an economy without potential for steady state growth, we assume a fixed supply of labor, λ > 0. Consumption possibilities are described
by
yi = Pi(li) (i = 1, 2) , (2)
l1+ l2 ≤ λ . (3)
In addition to the consumption good, each sector yields a pollutant which comes as a joint output in a fixed proportion to the desired output. Without loss of generality,
ei = yi (i = 1, 2) . (4)
Both flows of pollutants, e1 and e2, add to the respective stock of the pollutant,
which deteriorates at the constant rate δi:1
˙si= ei− δisi with δi > 0 (i = 1, 2) . (5)
Instantaneous social welfare V depends on consumption of both goods, y1 and
y2, and on the damage to environmental quality which hinges upon the stocks of
pollutants s1 and s2. We consider the following welfare function:
V (y1, y2, s1, s2) = U (y1, y2) − hσ1 2 s 2 1+ σ2 2 s 2 2 i with σ1, σ2> 0 , (6)
where σiindicates the harmfulness of pollutant i (i = 1, 2) and U represents welfare
gains due to consumption. The function U is assumed to exhibit the usual property of positive and decreasing marginal welfare in both consumption goods. In order to have an additively separable welfare function in all four arguments (y1, y2, s1, s2),
we assume that neither consumption good influences marginal welfare of the other. With index i denoting the partial derivative with respect to argument yi, i.e.
Ui≡ ∂U/∂yi and Uij ≡ ∂2U/∂yi∂yj with i, j = 1, 2, the assumptions are:
Ui > 0 , lim yi→0
Ui = +∞ , Uii< 0 , Uij = 0 (i, j = 1, 2 and i 6= j) . (7)
1In general, the decay rate may depend on emissions and the stock: δ
i = δi(ei, si). For analytical tractability, we assume δito be constant.
Both stocks of pollutants exert an increasing marginal damage, which is captured in the welfare function V , for the sake of tractability, by quadratic damage func-tions. Furthermore, both stocks decrease welfare independently. This is plausible if they damage different environmental goods. Thus, the welfare effect of one ad-ditional unit of one pollutant does not depend on the amount of the other. Note that the overall welfare function V is strictly concave.
Since we are interested in studying questions related to the scale as well as the structure of economic activity, and in order to simplify the analysis of corner solutions in the optimization problem, we introduce new dimensionless variables in the following way:
c = l1+ l2
λ and x =
l1
l1+ l2
. (8)
The variable c stands for the scale of economic activity. It indicates what fraction of the total available amount of labor is devoted to economic activity, and may take values between 0 and 1. The remaining fraction 1 − c is left idle. This can be interpreted as an implicit form of pollution abatement. By not using all available labor in the production of the consumption goods (and, consequently, emissions) but leaving part of the labor endowment idle, the variable c can be thought of as measuring the scale of economic activity in the sectors producing consumption goods and pollution, whereas the fraction 1 − c of labor may be thought of as being employed in (implicit) pollution abatement.2 The variable x stands for the structure of economic activity. It indicates the fraction of the total labor employed in production, l1+ l2, that is allocated to sector 1, and may take values between
0 and 1. The remaining fraction 1 − x is allocated to sector 2. The variables l1
2Not taking into account potential abatement activities for the scale of economic activity is in line with arguments from the ‘green national product’ discussion, according to which defensive and restorative activities should not be counted as augmenting the net national product (e.g. Ahmad et al. 1989, World Bank 1997).
and l2 can then be expressed in terms of c and x:
l1= l1(c, x) = cxλ and l2 = l2(c, x) = c(1 − x)λ .
This allows us to replace l1 and l2 in the problem. For notational convenience,
we introduce new production functions Fi which depend directly on c and x, and which are defined in the following way:
Fi(c, x) ≡ Pi(li(c, x)) for all c, x . (9)
From (1) and (9) one obtains that the Fi have the following properties:
Fc1 = xPl1λ > 0 , lim c→0F 1 c(x 6= 0) = +∞ , (10) Fx1 = cPl1λ > 0 , (11) Fc2 = (1 − x)Pl2λ > 0 , lim c→0F 2 c(x 6= 1) = +∞ , (12) Fx2 = −cPl2λ < 0 . (13)
3
Optimal scale and structure of the economy
Taking a social planner’s perspective, we now determine the optimal scale and structure of the multi-pollution economy described in the previous section. The control variables are the scale (c) and the structure (x) of economic activity. In terms of pollution, the choice over c and x is a choice over (i) how much pollution to emit overall, and (ii) what particular pollutant to emit. These are the two essential macroeconomic dimensions of every multi-pollution allocation decision.
3.1 Intertemporal optimization
We maximize the discounted intertemporal welfare over c and x, Z ∞ 0 h U (y1, y2) − σ1 2 s 2 1− σ2 2 s 2 2 i e−ρtdt , (14)
where ρ denotes the discount rate and yi = Fi(c, x) (i, j = 1, 2), subject to the
dynamic constraints for the two state variables s1 and s2 which are given by
Equations (5): ˙
si= Fi(c, x) − δisi with δi > 0 (i = 1, 2) . (15)
In addition, the following restrictions for the control variables c and x hold:
0 ≤ c ≤ 1 and 0 ≤ x ≤ 1 . (16)
Corner solutions with x = 0 or x = 1 cannot be optimal since either case would imply, due to Assumptions (1) and (7), that the marginal utility of one consump-tion good would go to infinity while the marginal utility of the other would remain finite. Similarly, a corner solution with c = 0 cannot be optimal since in that case the marginal utility of both consumption goods would go to infinity while the marginal damage from environmental pollution would remain finite. Hence, the only remaining restriction, which we have to control for explicitly, is:
c ≤ 1 . (17)
We introduce two costate variables, p1 and p2, and a Kuhn-Tucker parameter, pc.
The current value Hamiltonian of the problem then reads H(c, x, s1, s2; p1, p2, pc) = U (F1(c, x), F2(c, x)) − σ1 2 s 2 1− σ2 2 s 2 2 + p1F1(c, x) − δ1s1 + p2F2(c, x) − δ2s2 + pc[1 − c] . (18)
Since both control variables, c and x, are always strictly positive, the two state variables, s1and s2, are always nonnegative and the Hamiltonian H is continuously
problem are: U1Fc1 + U2Fc2 + p1Fc1 + p2Fc2 − pc = 0 , (19) U1Fx1 + U2Fx2 + p1Fx1 + p2Fx2 = 0 , (20) σ1s1 + (δ1+ ρ)p1 = ˙p1 , (21) σ2s2 + (δ2+ ρ)p2 = ˙p2 , (22) pc ≥ 0 , pc(1 − c) = 0 , (23)
plus the dynamic constraints (15) and the restriction (17). These necessary con-ditions are also sufficient if, in addition, the transversality concon-ditions
lim
t→∞pi(t) e −ρt· s
i(t) = 0 (i = 1, 2) , (24)
hold (see Appendix A.1). Note that the optimal path is also unique.
3.2 Stationary state
Setting ˙p1 = 0, ˙p2 = 0, ˙s1 = 0 and ˙s2 = 0 in the system of first order
condi-tions (15), (17) and (19)–(23) yields the necessary and sufficient condicondi-tions for an optimal stationary state (c⋆, x⋆, s⋆1, s⋆2), in which neither the scale nor the structure of economic activity nor the stocks of pollution accumulated in the environment change over time. From conditions (21) and (22) one obtains for the costate variables pi (i = 1, 2):
pi = −
σis⋆i
δi+ ρ
(i = 1, 2) . (25)
Inserting (25) in (19) and (20), and rearranging terms, yields the following neces-sary and sufficient conditions for an optimal stationary state:
U1⋆− σ1s ⋆ 1 δ1+ ρ = pcF 2 x ⋆ F1 c ⋆ F2 x ⋆ − F1 x ⋆ F2 c ⋆ , (26) U2⋆− σ2s ⋆ 2 δ2+ ρ = −pcF 1 x ⋆ F1 c ⋆ F2 x ⋆ − F1 x ⋆ F2 c ⋆ , (27)
where U⋆
i and Fji ⋆
(i = 1, 2 ; j = c, x) denote functions evaluated at stationary state values of the argument. From the signs of the Fji and pc stated in (10)–(13)
and (23), it follows that: Ui⋆ ≥ σis
⋆ i
δi+ ρ
(i = 1, 2) , (28)
where the “>” sign indicates a corner solution (c⋆ = 1). Furthermore, from the
equations of motion (15) one obtains s⋆i = F
i⋆
δi
= const. (i = 1, 2) . (29)
The interpretation of the two conditions (28) is that in an interior (corner) optimal stationary state the scale and structure of economic activity are such that for each sector the marginal welfare gain due to consumption of that sector’s output equals (is greater than) the aggregate future marginal damage from that sector’s current emission which comes as an inevitable by-product with the consumption good.3
An optimal stationary state exists if the system (23), (26), (27) and (29) of five equations for the five unknowns (c⋆, x⋆, s⋆
1, s⋆2) and p⋆c has a solution with
0 < c⋆ ≤ 1 and 0 < x⋆ < 1. With the properties of the utility and production functions assumed here, a unique optimal stationary state always exists.
Proposition 1:
(i) There exists a unique stationary state (c⋆, x⋆, s⋆
1, s⋆2), which is given as the
solution to (23), (26), (27) and (29).
(ii) The optimal stationary state of the economy is an interior solution with c⋆ < 1, if the total available amount of labor λ in the economy is strictly
greater than some threshold value ¯λ = ¯l1+ ¯l2, where the ¯li are specified by
3Note that taking account of discounting and the natural degradation of the respective pol-lution stock, the net present value of the accumulated damage of one marginal unit of polpol-lution sums up to the right-hand-side of (28), asR∞
0 σis ⋆ ie
−(ρ+δi)tdt = σ
the following implicit equations: Ui(P1(¯l1), P2(¯l2)) =
σiPi(¯li)
δi2+ δiρ
(i = 1, 2) .
Proof: see Appendix A.2.
In the following, we shall concentrate on the case of an interior stationary state with c⋆ < 1. Hence, we assume that the total labor amount λ exceeds ¯λ as specified
in Proposition 1. In order to study the properties of the interior optimal stationary state (c⋆, x⋆) some comparative statics can be done with Conditions (26), (27) and
(29). The results are stated in the following proposition.
Proposition 2:
An interior optimal stationary state, if it exists, has the following properties: dc⋆ dδ1 > 0 , dx ⋆ dδ1 > 0 , dc ⋆ dδ2 > 0 , dx ⋆ dδ2 < 0 , dc⋆ dσ1 < 0 , dx ⋆ dσ1 < 0 , dc ⋆ dσ2 < 0 , dx ⋆ dσ2 > 0 , dc⋆ dρ > 0 , dx⋆ dρ ≥ < 0 for [U⋆ 22δ2(δ2+ ρ) − σ2](δ2+ ρ) [U⋆ 11δ1(δ1+ ρ) − σ1](δ1+ ρ) ≥ < σ2F2⋆Fc1 ⋆ σ1F1⋆Fc2⋆ .
Proof: see Appendix A.3.
These results can be interpreted as follows. For both pollutants i (i = 1, 2), the lower is the natural deterioration rate δi and the higher is the harmfulness σi, the
lower is the relative weight of the emitting sector in the total economy and the lower is the overall scale of economic activity in the stationary state. An increase in the discount rate ρ increases the optimal stationary scale of economic activity, c⋆, while its effect on the optimal stationary structure of economic activity, x⋆, is
3.3 Optimal dynamic path and local stability analysis
In the following we solve the optimization problem by linearizing the resulting system of differential equations around the stationary state. Since our model is characterized by only mild non-linearities,4 we expect the linear approximation to
yield insights which should also hold for the exact problem. In Section 4 below, we shall numerically optimize the exact problem, and confirm this expectation.
As we have assumed an interior stationary state, the optimal path will also be an interior optimal path at least in a neighborhood of the interior stationary state. Hence, we restrict the analysis to the case of an interior solution, i.e. c⋆ < 1. As
shown in Appendix A.4, the optimal dynamics of the two control variables c, x and the two state variables s1, s2 can be described by a system of four coupled
first order autonomous differential equations: ˙c = [U1(δ1+ ρ) − σ1s1]U22F 2 x − [U2(δ2+ ρ) − σ2s2]U11Fx1 U11U22df , (30) ˙x = [U2(δ2+ ρ) − σ2s2]U11F 1 c − [U1(δ1+ ρ) − σ1s1]U22Fc2 U11U22df , (31) ˙s1 = F1− δ1s1 , (32) ˙s2 = F2− δ2s2 , (33)
with df ≡ Fc1Fx2−Fx1Fc2 < 0. Linearizing around the stationary state (c⋆, x⋆, s⋆ 1, s⋆2)
yields the following approximated dynamic system (see Appendix A.5): ˙c ˙x ˙ s1 ˙ s2 ≈ J⋆ c − c⋆ x − x⋆ s1− s⋆1 s2− s⋆2 with (34)
J⋆= ρ + δ1Fc1 ⋆ F2 x ⋆ −δ2Fx1 ⋆ F2 c ⋆ df⋆ (δ1−δ2)Fx1 ⋆ F2 x ⋆ df⋆ − σ1Fx2 ⋆ U⋆ 11df⋆ σ2Fx1 ⋆ U⋆ 22df⋆ (δ2−δ1)Fc1 ⋆ F2 c ⋆ df⋆ ρ + δ2Fc1 ⋆ F2 x ⋆ −δ1Fx1 ⋆ F2 c ⋆ df⋆ σ1Fc2 ⋆ U⋆ 11df⋆ − σ2Fc1 ⋆ U⋆ 22df⋆ Fc1⋆ Fx1⋆ −δ1 0 F2 c ⋆ F2 x ⋆ 0 −δ2 .
The Jacobian evaluated at the stationary state, J⋆, has four real eigenvalues (see
Appendix A.5), two of which are strictly negative (ν1, ν2) and two of which are
strictly positive (ν3, ν4). Hence, the system dynamics exhibits saddlepoint
sta-bility, i.e. for all initial stocks of pollutants, s01 and s02, there exists a unique optimal path which asymptotically converges towards the stationary state. Be-cause of the transversality conditions (24) the optimal path is restricted to the stable hyperplane, which is spanned by the eigenvectors associated with the nega-tive eigenvalues. Given the eigenvalues and the eigenvectors, which are calculated in Appendix A.5, the explicit system dynamics in a neighborhood around the stationary state is given by:
c(t) = c⋆+ (s01− s⋆1) F 2 x ⋆ (ν1+ δ1) F1 c⋆Fx2⋆− Fx1⋆Fc2⋆ eν1t− (s02− s⋆2) F 1 x ⋆ (ν2+ δ2) F1 c⋆Fx2⋆− Fx1⋆Fc2⋆ eν2t , (35) x(t) = x⋆− (s01− s⋆1) F 2 c ⋆ (ν1+ δ1) F1 c⋆Fx2⋆− Fx1⋆Fc2⋆ eν1t+ (s02− s⋆2) F 1 c ⋆ (ν2+ δ2) F1 c⋆Fx2⋆− Fx1⋆Fc2⋆ eν2t , (36) s1(t) = s⋆1+ (s01− s⋆1) eν1t , (37) s2(t) = s⋆2+ (s02− s⋆2) eν2t , (38) where s0
i = si(0) (i = 1, 2) denote the initial pollutant stocks.
As a measure of the overall rate of convergence of a process z(t) which asymp-totically approaches z⋆, we define the characteristic time scale of convergence τ
z by τz−1 ≡ ˙z(t) z(t) − z⋆ , (39)
where the horizontal bar denotes the average over time. The greater is the time scale τz, the slower is the convergence towards z⋆. With this definition, it is
obvi-ous from Equations (37) and (38) that the pollutant stock si (i = 1, 2) converges
towards its stationary state value s⋆
i with a characteristic time scale τsi = 1/ |νi|.
As the system approaches the stationary state for t → ∞, the scale c and struc-ture x (Equations 35 and 36) converge towards their stationary state values c⋆
and x⋆ with a characteristic time scale which is determined by the eigenvalue
with the smaller absolute value, τc = τx = 1/ min{|ν1|, |ν2|} (see Appendix A.6).
Proposition 3 summarizes these results. Proposition 3:
For the linear approximation (34) around the stationary state (c⋆, x⋆, s⋆
1, s⋆2) the
following statements hold:
(i) The stationary state is saddlepoint-stable.
(ii) The explicit system dynamics is given by Equations (35)–(38).
(iii) The characteristic time scale of convergence towards the stationary state is given by
• τc= τx= 1/ min{|ν1|, |ν2|} for the control variables c and x, and by
• τsi= 1/ |νi| for stock variable si (i = 1, 2).
As shown in Appendix A.5 the eigenvalues ν1 and ν2 are given by
ν1 = 1 2 " ρ − s (ρ + 2δ1)2−4σ1 U⋆ 11 # < 0 , (40) ν2 = 1 2 " ρ − s (ρ + 2δ2)2−4σ2 U⋆ 22 # < 0 . (41)
Hence, the absolute value of νi (time scale of convergence) decreases (increases)
with the discount rate ρ and the curvature of consumption welfare in the stationary state |U⋆
We now turn to the question of the (non-)monotonicity of the optimal path. According to Equations (37) and (38), the stocks of the two pollutants converge monotonically towards their stationary state values s⋆
1 and s⋆2. In order to show
that the optimal paths for the control variables c and x may be non-monotonic, we differentiate Equations (35) and (36) with respect to t:
˙c(t) = ν1(s01− s⋆1) F2 x ⋆ (ν1+ δ1) F1 c ⋆ F2 x ⋆ − F1 x ⋆ F2 c ⋆eν1t− ν2(s02− s⋆2) F1 x ⋆ (ν2+ δ2) F1 c ⋆ F2 x ⋆ − F1 x ⋆ F2 c ⋆eν2t , (42) ˙x(t) = −ν1(s01− s⋆1) F2 c ⋆ (ν1+ δ1) F1 c ⋆ F2 x ⋆ − F1 x ⋆ F2 c ⋆ eν1t+ ν2(s02− s⋆2) F1 c ⋆ (ν2+ δ2) F1 c ⋆ F2 x ⋆ − F1 x ⋆ F2 c ⋆eν2t . (43)
The optimal path is non-monotonic if ˙c or ˙x change their sign, i.e. if the paths c(t) or x(t) exhibit a local extremum for positive times t. According to the signs of the νi and Fji (i = 1, 2 and j = c, x) and given that ν1 6= ν2, c(t) exhibits a
unique local extremum if sgn(s01− s⋆
1) 6= sgn(s02− s⋆2), and x(t) exhibits a unique
local extremum if sgn(s01− s⋆1) = sgn(s02− s⋆2).5 Solving ˙c(t) = 0 and ˙x(t) = 0 for t, using expressions (42) and (43) for ˙c and ˙x, yields:
ˆ t = lnhν2(s02−s⋆2)Fx1 ⋆ (ν2+δ2) ν1(s01−s⋆1)Fx2⋆(ν1+δ1) i (ν1−ν2)−1 , if sgn(s01−s⋆1) 6= sgn(s02−s⋆2) lnhν2(s02−s⋆2)Fc1 ⋆ (ν2+δ2) ν1(s01−s⋆1)Fc2 ⋆ (ν1+δ1) i (ν1−ν2)−1 , if sgn(s01−s⋆1) = sgn(s02−s⋆2) . (44) According to this equation, it is possible that ˆt may be negative or infinite, which is meaningless in the context of this analysis. In this case we would observe monotonic optimal paths for both control variables c and x for times 0 < t < +∞. For instance, ˆt is negative if
s02− s⋆2
is sufficiently small, that is, the second pollutant stock is initially already close to its stationary state level. Furthermore, ˆ
t equals (plus or minus) infinity if either s0
1− s⋆1
= 0 or |ν1− ν2| = 0, that
5Note that ν
is, the first pollutant stock is initially already at its stationary state level or the eigenvalues are identical. The following proposition summarizes the behavior of the optimal control path.
Proposition 4:
In the linear approximation (34) around the stationary state (c⋆, x⋆, s⋆1, s⋆2), the following statements hold for the optimal path:
(i) The stocks of pollutants s1(t) and s2(t) converge exponentially, and hence
monotonically, towards their stationary state values s⋆
1 and s⋆2.
(ii) If and only if ˆt as given by Equation (44) is strictly positive and finite, then the optimal control is non-monotonic over time and ˆt denotes the time at which the optimal control has a unique local extremum. In particular, if sgn(s01− s⋆1) 6= sgn(s02− s⋆2), c(t) is non-monotonic and x(t) is monotonic. If sgn(s01− s⋆
1) = sgn(s02− s⋆2), x(t) is non-monotonic and c(t) is monotonic.
4
Numerical optimization
In this section we illustrate the results derived in Section 3 by numerical optimiza-tions of the original, non linearized optimization problem (14)–(16). The results thus obtained confirm that the insights from analyzing the linearized system also hold for the exact solution. All numerical optimizations were carried out with the advanced optimal control software package MUSCOD-II (Diehl et al. 2001), which exploits the multiple shooting state discretization (Leineweber et al. 2003).
There are four different qualitative scenarios which have to be examined. (i) Both stocks of pollutants exhibit the same harmfulness but differ in their deteriora-tion rates, i.e. σ1= σ2, δ1 < δ2. (ii) The two pollutants differ in their harmfulness
but have equal deterioration rates, i.e. σ1< σ2, δ1 = δ2. (iii) The pollutants differ
the higher deterioration rate, i.e. σ1 < σ2, δ1 < δ2. (iv) Both harmfulness and
deterioration rates are different, and the more harmful pollutant has a lower dete-rioration rate, i.e. σ1 < σ2, δ1> δ2. Furthermore, each of the four scenarios splits
into four subcases, depending on the initial stocks of pollutants (both initial stocks below, only first stock above, only second stock above and both stocks above the stationary state levels).
In the following we discuss these four different scenarios. The parameter values used for the numerical optimization have been chosen so as to illustrate clearly the different effects, and do not necessarily reflect the characteristics of real envi-ronmental pollution problems. For all numerical examples, the total labor supply λ has been chosen so as to guarantee an interior stationary state scale c⋆ < 1. As
it is not possible to optimize numerically over an infinite time horizon, the time horizon has been set to 250 years and all parameters have been chosen in such a way that the system at time t = 250 is very close to the stationary state. For a more convenient exposition, the figures show the time paths up to t = 125 only. The parameter values for the numerical optimization are listed in Appendix A.7. In the first scenario (σ1 = σ2), both stocks of pollutants exhibit the same
harmfulness but the deterioration rate is smaller for the first pollutant than for the second. Figure 1 shows the result of a numerical optimization of this case. In this example the initial stocks for both pollutants are above their stationary state levels (s0
1 = 30, s02 = 30). The optimal path for the structure exhibits non-monotonic
behavior as expected from Proposition 4. Further, we expect that the optimal stationary state structure x⋆ is clearly below 0.5, indicating that relatively more labor is employed in the second sector, because as the second stock of pollutant deteriorates at a higher rate the aggregate intertemporal damage of one unit of emissions is smaller for the second pollutant.6 This expectation is confirmed by
6Note that both consumption goods are equally valued by the representative consumer, i.e. µ1= µ2 (see Appendix A.7).
the numerical optimization. 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 scale structure t 0 5 10 15 20 25 30 35 0 20 40 60 80 100 120 stock1 stock2 t
Figure 1: Optimal paths for scale and structure (left) and the two pollutant stocks (right) for the case σ1 = σ2, δ1 < δ2. Parameter values used for the
numerical optimization are given in Appendix A.7.
In the second scenario (σ1 < σ2, δ1 = δ2), the two stocks of pollutants are of
different harmfulness but the deterioration rate for the two pollutants are equal. The result of a numerical optimization of this case is presented in Figure 2. In this example the initial stock for the first (second) pollutant is above (below) their stationary state levels (s1 = 40, s2 = 0). Now, the optimal path for the
scale exhibits a non monotonic behavior as expected from Proposition 4. Further, we expect that the optimal stationary state structure x⋆ is clearly above 0.5,
indicating that relatively more labor is employed by the second sector, because as the second stock of pollutant is less harmful the aggregate intertemporal damage of one unit of emissions is smaller for the second pollutant. This expectation is confirmed by the numerical optimization.
The third scenario (σ1 < σ2, δ1 < δ2) – both harmfulness and deterioration
rates are different and the more harmful pollutant has the higher deterioration rate – is the most interesting as neither of the two pollutants exhibits a priori more favorable dynamic characteristics for the economy. Hence, we are not able to predict which production sector will be used to a greater extent in the
station-0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 scale structure t 0 5 10 15 20 25 30 35 40 45 0 20 40 60 80 100 120 stock1 stock2 t
Figure 2: Optimal paths for scale and structure (left) and the two pollutant stocks (right) for the case σ1 < σ2, δ1 = δ2. Parameter values used for the
numerical optimization are given in Appendix A.7.
ary state. Furthermore, non monotonic paths – if they occur – are likely to be more pronounced than in the other cases. Figure 3 shows the optimal paths for a numerical example for all four subcases (initial pollutant stocks above or below stationary state levels for one and both pollutants). Of course, the long run sta-tionary state to which the economy converges, is the same in all four subscenarios, as all parameters are identical except for the initial stocks of the two pollutants. Nevertheless, the optimal paths and especially their convergence towards the sta-tionary state is quite different for the four subcases. As expected from Proposition 4, we observe that – if at all – the optimal path for the structure is non-monotonic if both stocks start above or below their stationary state levels (subcases a and d) and the optimal path for the scale is non-monotonic if one initial stock is higher and one is lower than their stationary state levels (subcase b). We also see that both, structure and scale, may exhibit monotonic optimal paths (subcase c).
In the fourth scenario (σ1< σ2, δ1> δ2), where both pollutants exhibit
differ-ent harmfulness and deterioration rates but the second pollutant is more harmful and has the lower deterioration rate, the first pollutant exhibits clearly more favor-able dynamic properties than the second pollutant. In this case the economy will
a) both stocks below stationary state level 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 scale structure t 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 stock1 stock2 t
b) first stock above, second stock below stationary state level
0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 scale structure t 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 stock1 stock2 t
c) first stock below, second stock above stationary state level
0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 scale structure t 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 stock1 stock2 t
d) both stocks above stationary state level
0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 scale structure t 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 stock1 stock2 t
Figure 3: Optimal paths for scale and structure (left) and the two pollutant stocks (right) for the case σ1 < σ2, δ1 < δ2 and all four subscenarios. Parameter
nearly exclusively use the first production sector. Although non-monotonicities in the optimal paths for scale and structure can occur according to Proposition 4, they are not pronounced. As nothing new can be learned from this case, we do not show a numerical optimization example.
5
Conclusion
In this paper, we have studied the mutual interaction over time between the scale and structure of economic activity on the one hand, and the dynamics of multiple environmental pollution stocks on the other hand. We have carried out a total analysis of a two-sector-economy, in which each sector produces one distinct con-sumption good and one specific pollutant. The pollutants of both sectors were assumed to differ in their environmental impact in two ways: (i) with respect to their harmfulness and (ii) with respect to their natural deterioration rates in the environment.
Most of the results are intuitive. First, it may be optimal not to use all avail-able labor endowment in the production of consumption goods in order to avoid excessive environmental damage. Second, under very general conditions a change in scale and structure of economic activity over time is optimal. Thus, the opti-mal economic dynamics is driven by the dynamics of the environmental pollution stocks. The less harmful is a pollutant, the higher are the relative importance of the emitting sector and the overall scale of economic activity in the stationary state. The shorter lived is a pollutant, the higher are the relative importance of the emitting sector and the overall scale of economic activity in the stationary state. If emissions differ either in their environmental harmfulness or in their de-terioration rates, we should have structural change towards the sector emitting the less harmful or the shorter-lived pollutant. However, if the harmfulness and deterioration rates differ and if the environmentally less harmful emission is also
the longer-lived pollutant, no general conclusion concerning the direction of struc-tural change can be drawn. Third, the characteristic time scale of convergence of scale and structure towards the stationary state is given by (the inverse of) the eigenvalue with the smaller absolute value. It increases with the discount rate and the curvature of consumption welfare in the stationary state; it decreases with the harmfulness and the deterioration rate of the respective pollutant stock.
Most importantly, our formal analysis as well as the numerical optimizations, show that it is likely that the optimal control paths, i.e. the change in the scale and structure of the economy, are non-monotonic over time.7 If a non-monotonic control is optimal, our numerical optimizations suggest that the local extremum of the control path may be pronounced and that it occurs at the beginning of the control path.
These results have implications for the design of environmental indicators and policies. First, the traditional view is that different environmental problems – such as e.g. acidification of soils and surface waters, groundwater contamination by nitrates or pesticides, and climate change due to anthropogenic greenhouse gas emission – can be regulated by independent environmental policies. In contrast, our total analysis of a multi-sector economy with several independent environ-mental pollutants, shows that these problems – even without any direct physical interaction – interact indirectly because they all affect social welfare, and the mit-igation of all of them is constrained by the available economic resources. As a result, even for non-interacting environmental pollutants the optimal regulation has to take an encompassing view, taking into account all of the environmental problems together.
7Non-monotonic optimal control paths, in particular limit-cycles, are known to exist for control problems with two or more state variables, and for time-lagged and adaptive control problems, even with one single state variable (e.g. Benhabib and Nishimura 1979, Feichtinger et al. 1994, Wirl 2000, 2002, Winkler 2004).
Second, indicators and policies which are solely based on the harmfulness of environmental pollutants – which is predominant in current environmental politics – fall short of optimally controlling environmental problems. In a dynamic set-ting, the lifetime of pollutants is an equally important determinant of the optimal environmental policy.
Third, the non-monotonicity-result challenges common intuition which sug-gests that policies should achieve optimal change in a monotonic way. In contrast to this simple intuition, our analysis shows that if pollutants accumulate on dif-ferent time scales and if they differ in environmental harmfulness, the optimal policies may be non-monotonic. In particular, the optimal time-path of structural change towards the stationary state structure may be characterized by ‘optimal overshooting’; that is, the optimal relative importance of a sector starts below (above) the stationary state level, increases (decreases) to a point above (below) the stationary state level, and finally decreases (increases) again. The same goes for the optimal dynamics of the overall economic scale.
Summing up, in order to develop sustainable solutions to the multiple environ-mental problems that we face in reality – such as climate change, depletion of the ozone layer, groundwater contamination, acidification of soil and surface water, biodiversity loss, etc. – we should adopt an encompassing view and base policy advice on a total analysis of economy-environment interactions. As our analysis shows, the resulting optimal policies need to take account of the history, the em-pirical parameter values and the dynamic relationships of all of the problems, and these policies might be non-monotonic.
Appendix
A.1 Concavity of the optimized Hamiltonian
We show that the Hamiltonian H, without taking into account the restriction c ≤ 1, i.e. pc = 0, is strictly concave whenever the necessary conditions are
satisfied. Thus, the unique optimal solution is the local extremum of H if we have an interior solution; it is a corner solution with c = 1 if the local extremum of H is reached for unfeasible c > 1.
A sufficient condition for strict concavity of the Hamiltonian is that its Hessian H = ∂2H
∂i∂j (i, j = c, x, s1, s2) is negative definite. The Hessian H reads:
H = Hcc Hcx 0 0 Hxc Hxx 0 0 0 0 −σ1 0 0 0 0 −σ2 (A.1)
Due to its diagonal form, H is negative definite if the reduced Hessian H′
=
∂2H
∂i∂j (i, j = c, x) is negative definite, i.e. Hcc, Hxx < 0 and det H ′
> 0.
Hcc = U11(Fc1)2+ (U1+ p1)Fcc1 + U22(Fc2)2+ (U2+ p2)Fcc2 , (A.2)
Hxx = U11(Fx1)2+ (U1+ p1)Fxx1 + U22(Fx2)2+ (U2+ p2)Fxx2 , (A.3)
Hcx = U11Fc1Fx1+ (U1+ p1)Fcx1 + U22Fc2Fx2+ (U2+ p2)Fcx2 . (A.4)
Along the optimal path, the necessary conditions have to be satisfied. In partic-ular, for an interior solution, i.e. c⋆ < 1, the necessary and sufficient conditions
(19) and (20) become:
(U1+ p1)Fc1+ (U2+ p2)Fc2 = 0 , (A.5)
(U1+ p1)Fx1+ (U2+ p2)Fx2 = 0 . (A.6)
With this, one obtains: Hcc = U11(Fc1)2+ U22(Fc2)2 < 0 , (A.8) Hxx = U11(Fx1)2+ U22(Fx2)2 < 0 , (A.9) det H′ = HccHxx− Hcx2 = U11U22(Fc1)2(Fx2)2+ (Fx1)2(Fc2)2− 2Fc1Fx1Fc2Fx2 > 0 . (A.10) Hence, whenever H has an extremum it is a maximum. As a consequence, the necessary conditions (plus the transversality condition 24) are also sufficient.
A.2 Proof of Proposition 1
(i) Inserting Equations (29) into Equations (26) and (27), and using the relation-ship between Fi and Pi, as given from Equation (9), one obtains:
Ui⋆ = σiP i⋆ δi(δi+ ρ) + pc λPi l ⋆ (i = 1, 2) . (A.11)
With the properties for Pi, as given by (1), and the properties for U
i, as given
by (7), the left-hand-side of Equation (A.11) is strictly decreasing while the right-hand-side is strictly increasing in li. Thus, there exists at most one li⋆ which
satisfies Equation (A.11). The existence of such a solution is guaranteed by the properties limli→0P
i
l = +∞ and limyi→0Ui= +∞.
(ii) We derive ¯li by solving (A.11) for l⋆i assuming pc = 0. Thus, ¯li is the maximal
amount of labor which will be assigned to production process i in an optimal stationary state without taking account for the restriction c ≤ 1. If ¯l1+ ¯l2 ≥ λ the
labor supply is short of the optimal labor demand and thus the stationary state is a corner solution. If, on the other hand, the total labor supply λ exceeds the sum ¯l1+ ¯l2, then not all labor will be used for economic activity and the optimal
A.3 Proof of Proposition 2
Setting pc = 0 in Equation (A.11) yields for an interior stationary path:
Ui⋆ = σiF
i⋆
δi(δi+ ρ)
(i = 1, 2) . (A.12)
By implicit differentiation of (A.12) with respect to δj (j = 1, 2) one obtains:
Fcj⋆∂c ⋆ ∂δj + Fxj⋆∂x ⋆ ∂δj Ujj⋆ − σj δj(δj + ρ) = −σjF j ⋆(2δ j+ ρ) δj2(δj+ ρ)2 (j = i) , Fci ⋆∂c⋆ ∂δj + Fxi ⋆∂x⋆ ∂δj Uii⋆− σi δi(δi+ ρ) = 0 (j 6= i) . Solving for ∂c⋆/∂δj and ∂x⋆/∂δj yields:
∂c⋆ ∂δj = σjF j ⋆Fi x ⋆ (2δj+ ρ) (Fi c ⋆ Fxj ⋆ − Fcj ⋆ Fi x ⋆ )(U⋆ jjδj(δj + ρ) − σj)δj(δj + ρ) , (A.13) ∂x⋆ ∂δj = σjF j ⋆Fi c ⋆ (2δj+ ρ) (Fcj ⋆ Fi x ⋆ − Fi c ⋆ Fxj ⋆ )(U⋆ jjδj(δj + ρ) − σj)δj(δj + ρ) . (A.14)
From the signs of the Fi
j (i = 1, 2; j = c, x) it follows that ∂c⋆ ∂δ1 > 0 , ∂c ⋆ ∂δ2 > 0, ∂x ⋆ ∂δ1 > 0, ∂x ⋆ ∂δ2 < 0 . (A.15)
By implicit differentiation of (A.12) with respect to σj (j = 1, 2) one obtains:
Fcj ⋆∂c⋆ ∂σj + Fxj ⋆∂x⋆ ∂σj Ujj⋆ − σj δj(δj + ρ) = F j ⋆ δj(δj + ρ) (j = i) , Fci⋆∂c ⋆ ∂σj + Fxi⋆∂x ⋆ ∂σj Uii⋆ − σi δi(δi+ ρ) = 0 (j 6= i) . Solving for ∂c⋆/∂σ j and ∂x⋆/∂σj yields: ∂c⋆ ∂σj = F j ⋆Fi x ⋆ (Fcj ⋆ Fi x ⋆ − Fi c ⋆ Fxj ⋆ )(U⋆ jjδj(δj + ρ) − σj) , (A.16) ∂x⋆ ∂σj = F j ⋆Fi c ⋆ (Fi c ⋆ Fxj ⋆ − Fcj ⋆ Fi x ⋆ )(U⋆ jjδj(δj + ρ) − σj) . (A.17)
From the signs of the Fi
j (i = 1, 2; j = c, x) it follows that ∂c⋆ ∂σ1 < 0 , ∂c ⋆ ∂σ2 < 0, ∂x ⋆ ∂σ1 < 0, ∂x ⋆ ∂σ2 > 0 . (A.18)
Implicit differentiation of (A.12) with respect to ρ yields: Fc1⋆∂c ⋆ ∂ρ + F 1 x ⋆∂x⋆ ∂ρ = − σ1F1⋆ [U⋆ 11δ1(δ1+ ρ) − σ1](δ1+ ρ) , Fc2⋆∂c ⋆ ∂ρ + F 2 x ⋆∂x⋆ ∂ρ = − σ2F2⋆ [U⋆ 22δ2(δ2+ ρ) − σ2](δ2+ ρ) , Solving for ∂c⋆/∂ρ and ∂x⋆/∂ρ yields:
∂c⋆ ∂ρ(F 2 c ⋆ Fx1⋆− Fc1⋆Fx2⋆) = = σ1F 1⋆F2 x ⋆ [U⋆ 11δ1(δ1+ ρ) − σ1](δ1+ ρ) − σ2F 2⋆F1 x ⋆ [U⋆ 22δ2(δ2+ ρ) − σ2](δ2+ ρ) , (A.19) ∂x⋆ ∂ρ (F 2 c ⋆ Fx1⋆− Fc1⋆Fx2⋆) = = σ2F 2⋆F1 c ⋆ [U⋆ 22δ2(δ2+ ρ) − σ2](δ2+ ρ) − σ1F 1⋆F2 c ⋆ [U⋆ 11δ1(δ1+ ρ) − σ1](δ1+ ρ) , (A.20) From the signs of the Fji (i = 1, 2; j = c, x) it follows that
∂c⋆ ∂ρ > 0 , ∂x⋆ ∂ρ ≥ < 0 ⇔ [U⋆ 22δ2(δ2+ ρ) − σ2](δ2+ ρ) [U⋆ 11δ1(δ1+ ρ) − σ1](δ1+ ρ) ≥ < σ2F2⋆Fc1 ⋆ σ1F1⋆Fc2 ⋆ . (A.21)
A.4 Derivation of the differential equation system
Differentiation of pi = −Ui (Equation A.7) with respect to time and inserting
into Equations (21) and (22) yields, together with the equations of motion (15), a system of four differential equations in the four unknowns c, x, s1 and s2:
σ1s1− U1(δ1+ ρ) + U11(Fc1˙c + Fx1˙x) = 0 , (A.22) σ2s2− U2(δ2+ ρ) + U22(Fc2˙c + Fx2˙x) = 0 , (A.23) ˙ s1− F1+ δ1s1 = 0 , (A.24) ˙ s2− F2+ δ2s2 = 0 . (A.25)
The conditions (A.22)–(A.25) for an interior optimal solution can be rearranged to yield the system (30)–(33) of four coupled autonomous differential equations.
A.5 Eigenvalues and eigenvectors of the Jacobian
We obtain the Jacobian J∗
by differentiating the right-hand-sides of Equations (30)–(33) with respect to c, x, s1 and s2 and evaluating them at the stationary
state. Taking into account that in the interior stationary state (28) holds with equality, Ui = σis⋆i/(δi+ ρ), one obtains for the Jacobian J⋆:
J∗= ρ + δ1Fc1 ⋆ Fx2 ⋆ −δ2Fx1 ⋆ Fc2 ⋆ df⋆ (δ1−δ2)Fx1 ⋆ Fx2 ⋆ df⋆ − σ1Fx2 ⋆ U⋆ 11df⋆ σ2Fx1 ⋆ U⋆ 22df⋆ (δ2−δ1)Fc1 ⋆ F2 c ⋆ df⋆ ρ + δ2Fc1 ⋆ F2 x ⋆− δ1Fx1 ⋆ F2 c ⋆ df⋆ σ1Fc2 ⋆ U⋆ 11df⋆ − σ2Fc1 ⋆ U⋆ 22df⋆ F1 c ⋆ F1 x ⋆ −δ1 0 Fc2 ⋆ Fx2 ⋆ 0 −δ2 .
The eigenvalues νiand eigenvectors ξiare the solutions of the equation J⋆·ξ = ν ·ξ.
The four eigenvalues are: ν1 = 1 2 " ρ − s (ρ + 2δ1)2− 4σ1 U⋆ 11 # < 0 , (A.26) ν2 = 1 2 " ρ − s (ρ + 2δ2)2− 4σ2 U⋆ 22 # < 0 , (A.27) ν3 = 1 2 " ρ + s (ρ + 2δ1)2− 4σ1 U⋆ 11 # > 0 , (A.28) ν4 = 1 2 " ρ + s (ρ + 2δ2)2− 4σ2 U⋆ 22 # > 0 . (A.29)
The eigenvectors associated with the negative eigenvalues ν1 and ν2 are:
ξ1 = F 2 x ⋆ (ν1+ δ1) df⋆ , − F2 c ⋆ (ν1+ δ1) df⋆ , 1, 0 , (A.30) ξ2 = −F 1 x ⋆ (ν2+ δ2) df⋆ , F1 c ⋆ (ν2+ δ2) df⋆ , 0, 1 . (A.31)
A.6 Time scale of convergence
Equations (35) and (36) are of the following type: z(t) = z⋆+ Aeν1t+ Beν2t (ν
1, ν2 < 0) , (A.32)
with real constants A and B. Without loss of generality assume that |ν1| <
|ν2|. Since we are interested in the system dynamics in a neighborhood of the
t → ∞. According to (39), the characteristic time scale of convergence of z in a neighborhood of the stationary state z⋆ is given by:
τ−1 z = lim t→∞ Aν1eν1t+ Bν2eν2t Aeν1t+ Beν2t = lim t→∞ Aν1+ Bν2e(ν2−ν1)t A + Be(ν2−ν1)t = |ν1| . (A.33)
Hence, for t → ∞ the characteristic time scale of convergence is constant and given by 1/ min{|ν1|, |ν2|}.
A.7 Parameter values for the numerical optimization
We used a Cobb-Douglas welfare function for the numerical optimizations,
U (y1, y2) = 0.5 ln(y1) + 0.5 ln(y2) , (A.34)
and the following production functions:
P1(l1) =pl1 , P2(l2) =pl2 . (A.35)
For all numerical optimizations we set λ = 1 and ρ = 0.03. In addition, we used the following parameter values for the different scenarios:
Figure σ1 σ2 δ1 δ2 s1 s2 1 0.01 0.01 0.02 0.1 30 30 2 0.003 0.03 0.05 0.05 40 0 3a 0.002 0.02 0.02 0.1 0 0 3b 0.002 0.02 0.02 0.1 50 0 3c 0.002 0.02 0.02 0.1 0 25 3d 0.002 0.02 0.02 0.1 50 25
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Working Paper Series in Economics
(see www.uni-lueneburg.de/vwl/papers for a complete list)
No.49 Helmut Fryges und Joachim Wagner: Exports and productivity growth – First evidence
from a continuous treatment approach, May 2007
No.48 Ulrich Kaiser und Joachim Wagner: Neue Möglichkeiten zur Nutzung vertraulicher
amtlicher Personen- und Firmendaten, April 2007
No.47 Joachim Wagner: Jobmotor Mittelstand? Arbeitsplatzdynamik und Betriebsgröße in der
westdeutschen Industrie, April 2007
No.46: Christiane Clemens und Maik Heinemann: Credit Constraints, Idiosyncratic Risks, and
the Wealth Distribution in a Heterogenous Agent Model, March 2007
No.45: Jan Kranich: Biotechnologie und Internationalisierung. Ergebnisse der
Online-Befragung, März 2007
No.44: Joachim Wagner: Entry, exit and productivity. Empirical results for German
manufacturing industries, March 2007
No.43: Joachim Wagner: Productivity and Size of the Export Market
Evidence for West and East German Plants, 2004, March 2007
No.42: Joachim Wagner: Why more West than East German firms export, March 2007
No.41: Joachim Wagner: Exports and Productivity in Germany, March 2007
No.40: Lena Koller, Klaus Schnabel und Joachim Wagner: Schwellenwerte im Arbeitsrecht.
Höhere Transparenz und Effizienz durch Vereinheitlichung, Februar 2007 [erscheint in: Perspektiven der Wirtschaftspolitik]
No.39: Thomas Wein und Wiebke B. Röber: Sind ausbildende Handwerksbetriebe
erfolgreicher?, Januar 2007
No.38: Joachim Wagner: Institut für Volkswirtschaft: Forschungsbericht 2006, Januar 2007
No.37: Nils Braakmann: The impact of September 11th, 2001 on the job prospects of foreigners
with Arab background – Evidence from German labor market data, January 2007
No.36: Jens Korunig: Regulierung des Netzmonopolisten durch Peak-load Pricing?, Dezember
2006
No.35: Nils Braakmann: Die Einführung der fachkundigen Stellungnahme bei der Ich-AG,
November 2006
No.34: Martin F. Quaas and Stefan Baumgärtner: Natural vs. financial insurance in the
management of public-good ecosystems, October 2006
No.33: Stefan Baumgärtner and Martin F. Quaas: The Private and Public Insurance Value of
Conservative Biodiversity Management, October 2006
No.32: Ingrid Ott and Christian Papilloud: Converging institutions. Shaping the relationships
between nanotechnologies, economy and society, October 2006
No.31: Claus Schnabel and Joachim Wagner: The persistent decline in unionization in western
and eastern Germany, 1980-2004: What can we learn from a decomposition analysis?, October 2006
[published in: Industrielle Beziehungen/The German Journal of Industrial Relations 14 (2007), 118-132]
No.29: Christian Growitsch and Heike Wetzel: Economies of Scope in European Railways: An
Efficiency Analysis, July 2006
No.28: Thorsten Schank, Claus Schnabel and Joachim Wagner: Do exporters really pay higher
wages? First evidence from German linked employer-employee data, June 2006 [published in in: Journal of International Economics 72 (2007), 1, 52-74]
No.27: Joachim Wagner: Markteintritte, Marktaustritte und Produktivität
Empirische Befunde zur Dynamik in der Industrie, März 2006 [erscheint in: Allgemeines Statistisches Archiv, Heft 3/2007]
No.26: Ingrid Ott and Susanne Soretz: Governmental activity and private capital adjustment,
March 2006
[forthcoming in: Icfai Journal of Managerial Economics]
No.25: Joachim Wagner: International Firm Activities and Innovation:
Evidence from Knowledge Production Functions for German Firms, March 2006
No.24: Ingrid Ott und Susanne Soretz: Nachhaltige Entwicklung durch endogene
Umweltwahrnehmung, März 2006
publiziert in: Clemens, C., Heinemann, M. & Soretz, S., Auf allen Märkten zu Hause (Gedenkschrift für Franz Haslinger), Marburg: Metropolis, 2006, 233-256
No.23: John T. Addison, Claus Schnabel, and Joachim Wagner: The (Parlous) State of German
Unions, February 2006
[forthcoming in: Journal of Labor Research 28 (2007), 3-18]
No.22: Joachim Wagner, Thorsten Schank, Claus Schnabel, and John T. Addison: Works
Councils, Labor Productivity and Plant Heterogeneity: First Evidence from Quantile Regressions, February 2006
[published in: Jahrbücher für Nationalökonomie und Statistik 226 (2006), 505 - 518]
No.21: Corinna Bunk: Betriebliche Mitbestimmung vier Jahre nach der Reform des BetrVG:
Ergebnisse der 2. Befragung der Mitglieder des Arbeitgeberverbandes Lüneburg Nordostniedersachsen, Februar 2006
No.20: Jan Kranich: The Strength of Vertical Linkages, July 2006
No.19: Jan Kranich und Ingrid Ott: Geographische Restrukturierung internationaler
Wertschöpfungsketten – Standortentscheidungen von KMU aus regionalökonomischer Perspektive, Februar 2006
No.18: Thomas Wein und Wiebke B. Röber: Handwerksreform 2004 – Rückwirkungen auf das
Ausbildungsverhalten Lüneburger Handwerksbetriebe?, Februar 2006
No.17: Wiebke B. Röber und Thomas Wein: Mehr Wettbewerb im Handwerk durch die
Handwerksreform?, Februar 2006
No.16: Joachim Wagner: Politikrelevante Folgerungen aus Analysen mit wirtschaftsstatistischen
Einzeldaten der Amtlichen Statistik, Februar 2006 [publiziert in: Schmollers Jahrbuch 126 (2006) 359-374]
No.15: Joachim Wagner: Firmenalter und Firmenperformance
Empirische Befunde zu Unterschieden zwischen jungen und alten Firmen in Deutschland, September 2005
[publiziert in: Lutz Bellmann und Joachim Wagner (Hrsg.), Betriebsdemographie (Beiträge zur Arbeitsmarkt- und Berufsforschung, Band 305), Nürnberg: IAB der BA,
No.14: Joachim Wagner: German Works Councils and Productivity:
First Evidence from a Nonparametric Test, September 2005 [forthcoming in: Applied Economics Letters]
No.13: Lena Koller, Claus Schnabel und Joachim Wagner: Arbeitsrechtliche Schwellenwerte und
betriebliche Arbeitsplatzdynamik: Eine empirische Untersuchung am Beispiel des Schwerbehindertengesetzes, August 2005
[publiziert in: Zeitschrift für ArbeitsmarktForschung/ Journal for Labour Market Research 39 (2006), 181-199]
No.12: Claus Schnabel and Joachim Wagner: Who are the workers who never joined a union?
Empirical evidence from Germany, July 2005
[published in: Industrielle Beziehungen/ The German Journal of Industrial Relations 13 (2006), 118-131]
No.11: Joachim Wagner: Exporte und Produktivität in mittelständischen Betrieben
Befunde aus der niedersächsischen Industrie (1995 – 2004), June 2005 [publiziert in: Niedersächsisches Landesamt für Statistik, Statistische Berichte
Niedersachsen, Sonderausgabe: Tagung der NLS am 9. März 2006, Globalisierung und regionale Wirtschaftsentwicklung - Datenlage und Datenbedarf in Niedersachsen. Hannover, Niedersächsisches Landesamt für Statistik, Juli 2006, 18 – 29]
No.10: Joachim Wagner: Der Noth gehorchend, nicht dem eignen Trieb.
Nascent Necessity and Opportunity Entrepreneurs in Germany.
Evidence from the Regional Entrepreneurship Monitor (REM), May 2005 [published in: RWI: Mitteilungen. Quarterly 54/ 55 (2003/04), 287-303 {published June 2006}]
No. 9: Gabriel Desgranges and Maik Heinemann: Strongly Rational Expectations Equilibria with
Endogenous Acquisition of Information, March 2005
No. 8: Joachim Wagner: Exports, Foreign Direct Investment, and Productivity: Evidence from
German Firm Level Data, March 2005
[published in: Applied Economics Letters 13 (2006), 347-349]
No. 7: Thomas Wein: Associations’ Agreement and the Interest of the Network Suppliers – The
Strategic Use of Structural Features, March 2005
No. 6: Christiane Clemens and Maik Heinemann: On the Effects of Redistribution on Growth
and Entrepreneurial Risk-Taking, March 2005
No. 5: Christiane Clemens and Maik Heinemann: Endogenous Redistributive Cycles – An
overlapping Generations Approach to Social Conflict and Cyclical Growth, March 2005
No. 4: Joachim Wagner: Exports and Productivity: A Survey of the Evidence from Firm Level
Data, March 2005
[published in: The World Economy 30 (2007), 1, 60-82]
No. 3: Thomas Wein and Reimund Schwarze: Is the Market Classification of Risk Always
Efficient? - Evidence from German Third Party Motor Insurance, March 2005
No. 2: Ingrid Ott and Stephen J. Turnovsky: Excludable and Non-Excludable Public Inputs:
Consequences for Economic Growth, June 2005 (Revised version) [published in: Economica 73 (2006), 292, 725-742
No. 1: Joachim Wagner: Nascent and Infant Entrepreneurs in Germany.
Evidence from the Regional Entrepreneurship Monitor (REM), March 2005 [published in: Simon C. Parker (Ed.), The Life Cycle of Entrepreneurial Ventures
(International Handbook Series on Entrepreneurship, Volume 3), New York etc.: Springer, 2006, 15-37]
Universität Lüneburg Institut für Volkswirtschaftslehre Postfach 2440 D-21314 Lüneburg Tel.: ++49 4131 677 2321 email: brodt@uni-lueneburg,de www.uni-lueneburg.de/vwl/papers
